Fast Multigrid Solvers for Higher Order Upwind Discretizations of Convection-Dominated Problems

1998 ◽  
pp. 212-224
Author(s):  
C. W. Oosterlee ◽  
F. J. Gaspar ◽  
T. Washio ◽  
R. Wienands
Keyword(s):  
Higher Order ◽  
Multigrid Solvers ◽  
Convection Dominated

Error estimates for higher-order finite volume schemes for convection–diffusion problems

2018 ◽  
Vol 26 (1) ◽  
pp. 35-62
Author(s):  
Dietmar Kröner ◽  
Mirko Rokyta
Keyword(s):  
Error Estimates ◽  
Finite Volume ◽  
Original Problem ◽  
A Priori ◽  
Higher Order ◽  
Finite Volume Schemes ◽  
Convection Diffusion ◽  
Type Reconstruction ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIt is still an open problem to provea priorierror estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domainΩin ℝ2and we can prove such kind of ana priorierror estimate. In the part of the estimate, which refers to the discretization of the convective term, we gainh1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.

The Influence of Higher Order FEM Discretisations on Multigrid Convergence

2006 ◽  
Vol 6 (2) ◽  
pp. 221-232 ◽  
Author(s):  
M. Köster ◽  
S. Turek
Keyword(s):  
Finite Elements ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Higher Order ◽  
Iterative Solvers ◽  
Elliptic Pdes ◽  
Element Approximation ◽  
Approximation Properties ◽  
Transfer Operators ◽  
Multigrid Solvers

AbstractQuadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.

SUCCA3D—An Alternative Scheme to Reduce False Diffusion in Three-Dimensional Flows

1993 ◽  
Vol 207 (5) ◽  
pp. 307-313 ◽  
Author(s):  
T J Scanlon ◽  
C Carey ◽  
S M Fraser
Keyword(s):  
Three Dimensional ◽  
Higher Order ◽  
Upwind Scheme ◽  
Scalar Transport ◽  
Benchmark Test ◽  
Alternative Scheme ◽  
First Order ◽  
Quick Scheme ◽  
Scalar Variable ◽  
Convection Dominated

An alternative flow-oriented convection algorithm is presented which acts as a replacement for the first-order accurate UPWIND scheme in three-dimensional scalar transport. The scheme, formally titled SUCCA3D (skew upwind corner convection algorithm 3D), attempts to follow local streamlines, thus directly reducing the multi-dimensional false diffusion of the conserved scalar. In a standard benchmark test of pure convection across a three-dimensional cavity the SUCCA3D scheme was found to compare favourably with alternative schemes such as UPWIND and the higher-order QUICK scheme. The results highlight the potential of the SUCCA3D code for the reduction of three-dimensional false diffusion of a scalar variable in convection-dominated flows.

A Compact Higher-Order Scheme for Two-Dimensional Unsteady Convection–Diffusion Equations

2019 ◽  
Vol 17 (07) ◽  
pp. 1950025
Author(s):  
Yon-Chol Kim
Keyword(s):  
Stability Analysis ◽  
Numerical Study ◽  
Higher Order ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Convection Diffusion ◽  
Order Scheme ◽  
Higher Order Scheme ◽  
Convection Dominated ◽  
Diffusion Problems

In this paper, we study a compact higher-order scheme for the two-dimensional unsteady convection–diffusion problems using the nearly analytic discrete method (NADM), especially, focusing on the convection dominated-diffusion problems. The numerical scheme is constructed and the stability analysis is implemented. We find the order of accuracy of scheme is [Formula: see text], where [Formula: see text] is the size of time steps and [Formula: see text] the size of spacial steps, especially, making clear the relation between [Formula: see text] and [Formula: see text] is according to the different values of diffusion parameter [Formula: see text] through von Neumann stability analysis. The obtained numerical results for several benchmark problems show that our method makes progress in the numerical study of NADM for convection–diffusion equation and is to be effective and helpful particularly in computations for the convection dominated-diffusion equations and, furthermore, valuable in the numerical treatment of many real-world problems such as MHD natural convection flow.

Multigrid Line Smoothers for Higher Order Upwind Discretizations of Convection-Dominated Problems

1998 ◽  
Vol 139 (2) ◽  
pp. 274-307 ◽  
Author(s):  
C.W. Oosterlee ◽  
F.J. Gaspar ◽  
T. Washio ◽  
R. Wienands
Keyword(s):  
Higher Order ◽  
Convection Dominated
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